# Expected number of days inside a corridor

Is there a simple (ish) approximation for the expected number of steps a random walk is within a set of bounds over a given time period? - in particular if i presume log normal and constant vol.

If i want to do it with something like local/stochastic/both vol, then is there a way to do it without resorting to MC/diffusion?

• Can you remind us what is the payoff of a corridor? – Gordon Jun 28 '16 at 15:14
• The way I interpret this is the Stopping Time $\tau$ until $x(\tau) \geq B1 ∨ x(\tau) \leq B2$ – noob2 Jun 28 '16 at 15:29
• Sorry, should have been clearer - it's the total number of days it is within the corridor, it does not knock out. – will Jun 28 '16 at 15:50

This looks to me like a range accrual. Let $t_1, \ldots, t_n$, where $0 < t_1 < \cdots < t_n$ be business days that are being considered. We compute \begin{align*} E\left(\sum_{i=1}^n \pmb{1}_{b_1 < S_{t_i} < b_2} \right) &=\sum_{i=1}^n E\left(\pmb{1}_{b_1 < S_{t_i} < b_2} \right)\\ &=\sum_{i=1}^n \left[E\big(\pmb{1}_{S_{t_i} > b_1}\big) -E\big(\pmb{1}_{S_{t_i} \ge b_2} \big) \right]\\ &=\sum_{i=1}^n \left[\Phi\big(d_2^i(b_1)\big) - \Phi\big(d_2^i(b_2)\big)\right], \end{align*} where, for positive constant $K$, \begin{align*} d_2^i(K) = \frac{\ln\frac{S_0}{K} + \big(r-\frac{1}{2}\sigma^2 \big) t_i}{\sigma \sqrt{t_i}}. \end{align*} For general stochastic volatility, there is no simple method. Monte Carlo may be needed.